\documentclass[10pt,notitlepage]{article}
\usepackage{amsmath,graphicx,amssymb,datetime,stmaryrd,txfonts,dbnsymb,amscd}
\usepackage[setpagesize=false]{hyperref}
\usepackage[all]{xy}
\usepackage[usenames,dvipsnames]{xcolor}
% Following http://tex.stackexchange.com/a/847/22475:
\usepackage[setpagesize=false]{hyperref}\hypersetup{colorlinks,
  linkcolor={blue!50!black},
  citecolor={blue!50!black},
  urlcolor=blue
}

\paperwidth 8.5in
\paperheight 11in
\textwidth 8.5in
\textheight 11in
\oddsidemargin -1in
\evensidemargin \oddsidemargin
\topmargin -1in
\headheight 0in
\headsep 0in
\footskip 0in
\parindent 0in
\setlength{\topsep}{0pt}
\pagestyle{empty}

\def\blue{\color{blue}}
\def\green{\color{green}}
\def\red{\color{red}}

\def\bbC{{\mathbb C}}
\def\bbR{{\mathbb R}}
\def\calA{{\mathcal A}}
\def\calC{{\mathcal C}}
\def\calD{{\mathcal D}}
\def\calE{{\mathcal E}}
\def\calK{{\mathcal K}}
\def\calU{{\mathcal U}}
\def\frakg{{\mathfrak g}}

\def\Aut{{\operatorname{Aut}}}
\def\gr{{\operatorname{gr}}}
\def\hol{{\operatorname{hol}}}
\def\tr{{\operatorname{tr}}}

\def\act{\!\sslash\!}
\def\arXiv#1{{\href{http://front.math.ucdavis.edu/#1}{arXiv:\linebreak[0]#1}}}
\def\qed{{\hfill\text{$\Box$}}}

\def\navigator{{Dror Bar-Natan: Talks: Louvain-1506:}}

\def\w#1{{\href{http://www.math.toronto.edu/drorbn/Talks/Louvain-1506/#1}{$\omega$/#1}}}
\def\webdef{{$\omega\coloneqq$\url{http:drorbn.net/Louvain-1506}}}
\def\webnote{{Handout, video, and links at \w{}}}

\def\Recall{{\raisebox{2mm}{\parbox[t]{3.75in}{
{\red Recall.} $\calK=\{\text{knots}\}$, $\calA\coloneqq\gr\calA=\calD/\text{rels}=$
\vskip 1in
Seek $Z\colon\calK\to\hat\calA$ such that if $K$ is $n$-singular, $Z(K)=D_k+\ldots$
\[ \begin{CD}
  \calK
  @>\text{\color{red}$Z$: high algebra}>\parbox{1in}{\scriptsize
    solving finitely many equations in finitely many unknowns
  }>
  {\calA\coloneqq\gr\calK}
  @>\parbox{0.8in}{\centering\scriptsize
    given a ``Lie'' algebra $\frakg$
  }>\parbox{0.8in}{\scriptsize
    {\color{red}low algebra:} pictures represent formulas
  }>
  \text{``$\calU(\frakg)$''}
\end{CD} \]
}}}}

\def\Theorem{{\raisebox{2mm}{\parbox[t]{3.75in}{
{\red Theorem.} Given a parametrized knot $\gamma$ in $\bbR^3$, up to renormalizing the ``framing
anomaly'',

\[ Z(\gamma)=\sum_{D\in\calD}\frac{C(D)D}{|\Aut(D)|}\int_{\calC_D(\bbR^3,\gamma)}
  \bigwedge_{e\in E(D)}\phi_e^\ast\omega
  \in\calA
\]

\vskip 1mm
is an expansion. Here $\calD$ is the set of all ``Feynman diagrams'', $E(D)$ is the set of internal
edges (and chords) of $D$, $\calC_D(\bbR^3,\gamma)$ is the
configuration space of placements of $D$ on/around $\gamma$,
$\phi\colon\calC_D(\bbR^3,\gamma)\to(S^2)^{E(D)}$ is the ``direction of the edges'' map, and $\omega$
is a volume form on $S^2$.
}}}}

\def\DK{{$\displaystyle
  \langle D,K\rangle_\stonehenge:=\left(\parbox{100pt}{
    The signed Stonehenge pairing of $D$ and $K$
  }\right):
$}}

\def\Zdef{{$\displaystyle
  Z(K) := \lim_{N\to\infty}
    \sum_{\mbox{\scriptsize 3-valent }D}
    \frac{\langle D,K\rangle_\stonehenge D}{2^c c!{N\choose e}}
    \in\calA
$}}

\def\Fourier{{\raisebox{2mm}{\parbox[t]{2.5in}{
{\red The Fourier Transform.}
\[ (F\colon V\to\bbC)\Rightarrow(\tilde{f}\colon V^\ast\to\bbC) \]
via $\tilde{F}(\varphi)\coloneqq\int_Vf(v)e^{-i\langle\varphi,v\rangle}dv$.
Some facts:

\vskip 1mm
$\bullet$ $\tilde{f}(0)=\int_Vf(v)dv$.

\vskip 1mm
$\bullet$ $\frac{\partial}{\partial\varphi_i}\tilde{f}\sim\widetilde{v^if}$.

\vskip 1mm
$\bullet$ $\widetilde{(e^{Q/2})}\sim e^{Q^{-1}/2}$, where $Q$ is quadratic,
$Q(v)=\langle Lv,v\rangle$ for $L\colon V\to V^\ast$, and
$Q^{-1}(\varphi)\coloneqq\langle\varphi,L^{-1}\varphi\rangle$. (This is the
key point in the proof of the Fourier inversion formula!)
}}}}

\def\Monsters{{\raisebox{2mm}{\parbox[t]{2.5in}{
{\red Monsters left to Slay.}
\newline$\bullet$ Convergence.
\newline$\bullet$ Proof of invariance.
\newline$\bullet$ The framing anomaly.
\newline$\bullet$ Universallity.
\newline$\bullet$ $d^{-1}$ doesn't really exist, Faddeev-Popov, determinants, ghosts, Berezin
integration.
\newline$\bullet$ Assembly.
}}}}

\def\ChernSimons{{\raisebox{2mm}{\parbox[t]{4.2in}{
\parshape 3 0in 3.65in 0in 3.65in 0in 4.2in
{\red Claim.} It all comes from the Chern-Simons-Witten theory,
\newline\null\ \ $\displaystyle
  \int_{A\in\Omega^1(\bbR^3,\frakg)}
  \hspace{-30pt} \calD A\,
  \tr_R\mbox{\it hol}_\gamma(A)\exp\left[
    \frac{ik}{4\pi}\int\limits_{\bbR^3}
    \mbox{tr}\left(A\wedge dA+\frac{2}{3}A\wedge A\wedge A\right)
  \right],
$
where $\Omega^1(\bbR^3,\frakg)$ is the space of all $\frakg$-valued 1-forms on $\bbR^3$ (really,
connections), $k$ is some large constant, $R$ is some representation of $\frakg$ and $\tr_R$ is trace
in $R$, and $\mbox{\it hol}_\gamma(A)$ is the holonomy of $A$ along $\gamma$.
}}}}

\def\References{{\raisebox{2mm}{\parbox[t]{4.2in}{
{\red References.} Witten's {\em Quantum field theory and the Jones
polynomial}, Axelrod-Singer's {\em Chern-Simons perturbation theory
I-II}, D.~Thurston's \arXiv{math.QA/9901110}, Polyak's
\arXiv{math.GT/0406251}, and my videotaped 2014 class \w{AKT}.
}}}}

\def\main{{\raisebox{9pt}{\parbox[t]{5.4in}{
{\red Gaussian Integration.} $(\lambda_{ij})$ is a symmetic positive definite
matrix and $(\lambda^{ij})$ is its inverse, and $(\lambda_{ijk})$ are the
coefficients of some cubic form. Denote by $(x^i)_{i=1}^n$ the
coordinates of $\bbR^n$, let $(t_i)_{i=1}^n$ be a set of ``dual''
variables, and let $\partial^i$ denote $\frac{\partial}{\partial t_i}$.
Also let $C\coloneqq\frac{(2\pi)^{n/2}}{\det(\lambda_{ij})}$. Then
\begin{multline*}
  \int\limits_{\bbR^n}
    e^{-\frac12\lambda_{ij}x^ix^j+\frac{\epsilon}{6}\lambda_{ijk}x^ix^jx^k} 
  = \sum_{m\geq 0}\frac{\epsilon^m}{6^mm!}
    \int\limits_{\bbR^n}
    (\lambda_{ijk}x^ix^jx^k)^m e^{-\frac12\lambda_{ij}x^ix^j} \\
  = \left. \sum_{m\geq 0}\frac{C\epsilon^m}{6^mm!}
      (\lambda_{ijk}\partial^i\partial^j\partial^k)^m
      e^{\frac12\lambda^{\alpha\beta}t_\alpha t_\beta}
    \right|_{t_\alpha=0}
  = \sum_{\substack{m,l\geq 0 \\ 3m=2l }} \frac{C\epsilon^m}{6^mm!2^ll!}
      \left(\lambda_{ijk}\partial^i\partial^j\partial^k\right)^m
      \left(\lambda^{\alpha\beta}t_\alpha t_\beta\right)^l \\
  = \sum_{\substack{m,l\geq 0 \\ 3m=2l }} \frac{C\epsilon^m}{6^mm!2^ll!}
    \left[\begin{array}{c}\input{Pitchforks.pstex_t}\end{array}\right] \\
  = \sum_{\substack{m,l\geq 0 \\ 3m=2l }} \frac{C\epsilon^m}{6^mm!2^ll!}
    \sum_{\substack{m\text{-vertex fully marked} \\ \text{Feynman diagrams }D}}
      \hspace{-18pt}\calE(D)
      \def\Da{$\lambda_{i_1j_1k_1}\lambda_{i_2j_2k_2}
        \lambda^{i_1i_2}\lambda^{j_1j_2}\lambda^{k_1k_2}$}
      \def\Db{$\lambda_{i_1j_1k_1}\lambda_{i_2j_2k_2}
        \lambda^{i_1j_1}\lambda^{k_1k_2}\lambda^{i_2j_2}$}
      \qquad\begin{array}{c}\input{MarkedDiagrams.pstex_t}\end{array} \\
  = C\sum_{\substack{\text{unmarked Feynman} \\ \text{diagrams }D}}
      \frac{\epsilon^{m(D)}\calE(D)}{|\Aut(D)|}.
    \qquad\parbox[t]{3in}{
      {\bf Claim.} The number of pairings that produce a given unmarked
      Feynman diagram $D$ is $\frac{6^mm!2^ll!}{|\Aut(D)|}$.
    }
\end{multline*}
{\bf Proof of the Claim.} The group $G_{m,l}\coloneqq[(S_3)^m\rtimes
S_m]\times[(S_2)^l\rtimes S_l]$ acts on the set of pairings, the action
is transitive on the set of pairings $P$ that produce a given $D$,
and the stabilizer of any given $P$ is $\Aut(D)$. \qed
}}}}

\begin{document}
\setlength{\jot}{3pt}
\setlength{\abovedisplayskip}{0ex}
\setlength{\belowdisplayskip}{0ex}
\setlength{\abovedisplayshortskip}{0ex}
\setlength{\belowdisplayshortskip}{0ex}

\begin{center}
\null\vfill
\input{Louvain3.pstex_t}
\vfill\null
\end{center}

\end{document}

\endinput